We consider three matrix models of order 2 with one random entry $ \epsilon $ and the other three entries being deterministic. In the first model, we let $ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when $ \epsilon\sim \rm{Bernoulli}\left(p\right) $ and $ p\in [0, 1] $ is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with $\xi\epsilon$ where $\epsilon$ is a standard Cauchy random variable and $\xi$ is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.
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